Moving /var/lib/mysql to /home/mysql/

 

With MySQL not running, you should be able to simply create /home/mysql and copy across all the files using:
sudo su -
cp -a /var/lib/mysql/* /home/mysql/

BUT you must almost update AppArmor:
sudo vim /etc/apparmor.d/usr.sbin.mysql

And add the following rules:
/home/mysql/ rwk,
/home/mysql/** rwk,

Then run:
sudo apparmor_parser -r /etc/apparmor.d/usr.sbin.mysqld

Now start MySQL